|
|
|
|
Elliptic geometry
|
| |
|
| |
Elliptic geometry (sometimes known as Riemannian geometry) is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p. Elliptic geometry, like hyperbolic geometry, violates Euclid's parallel postulate, which can be interpreted as asserting that there is exactly one line parallel to L passing through p. In elliptic geometry, there are no parallel lines at all.
Discussion
Ask a question about 'Elliptic geometry' Start a new discussion about 'Elliptic geometry' Answer questions from other users |
Encyclopedia
Elliptic geometry (sometimes known as Riemannian geometry) is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p. Elliptic geometry, like hyperbolic geometry, violates Euclid's parallel postulate, which can be interpreted as asserting that there is exactly one line parallel to L passing through p. In elliptic geometry, there are no parallel lines at all. Elliptic geometry has a variety of properties that differ from those of classical Euclidean plane geometry. For example, the sum of the angles of any triangle is always greater than 180°.
Two dimensions
The spherical model
A simple way to picture elliptical geometry is to look at a globe. Neighboring lines of longitude appear to be parallel at the equator, yet they intersect at the poles.
More precisely, the surface of a sphere is a model of elliptic geometry if lines are modeled by great circles, and points at each other's antipodes are considered to be the same point. With this identification of antipodal points, the model satisfies Euclid's first postulate, which states that two points uniquely determine a line. If the antipodal points were considered to be distinct, as in spherical geometry, then uniqueness would be violated, e.g., the lines of longitude on the Earth's surface all pass through both the north pole and the south pole.
Although models such as the spherical model are useful for visualization and for proof of the theory's self-consistency, neither a model nor an embedding in a higher-dimensional space is logically necessary. For example, Einstein's theory of general relativity has static solutions in which space containing a gravitational field is (locally) described by three-dimensional elliptic geometry, but the theory does not posit the existence of a fourth spatial dimension, or even suggest any way in which the existence of a higher-dimensional space could be detected. (This is unrelated to the treatment of time as a fourth dimension in relativity.) Metaphorically, we can imagine geometers who are like ants living on the surface of a sphere. Even if the ants are unable to move off of the surface, they can still construct lines and verify that parallels do not exist. The existence of a third dimension is irrelevant to the ants' ability to do geometry, and its existence is neither verifiable nor necessary from their point of view. Another way of putting this is that the language of the theory's axioms is incapable of expressing the distinction between one model and another.
Comparison with Euclidean geometry
In Euclidean geometry, a figure can be scaled up or scaled down indefinitely, and the resulting figures are similar, i.e., they have the same angles and the same internal proportions. In elliptic geometry this is not the case. For example, in the spherical model we can see that the distance between any two points must be strictly less than half the circumference of the sphere (because antipodal points are identified). A line segment therefore cannot be scaled up indefinitely. A geometer measuring the geometrical properties of the space he or she inhabits can detect, via measurements, that there is a certain distance scale that is a property of the space. On scales much smaller than this one, the space is approximately flat, geometry is approximately Euclidean, and figures can be scaled up and down while remaining approximately similar.
A great deal of Euclidean geometry carries over directly to elliptic geometry. For example, the first 28 propositions in book I of Euclid's elements are proved without the use of the parallel postulate, and are therefore still valid in elliptic geometry, provided that some of the other postulates are given appropriate interpretations. For example, postulate 3, to construct a circle with any given center and radius, fails if "any radius" is taken to mean "any real number," but holds if it is taken to mean "the length of any given line segment." An example of a result that holds in both Euclidean and elliptic geometry is proposition 1 from book I of the Elements, which states that given any line segment, an equilateral triangle can be constructed with the segment as its base. Such propositions fall within the scope of absolute geometry.
Elliptic geometry is also like Euclidean geometry in that space is continuous, homogeneous, isotropic, and without boundaries. Isotropy is guaranteed by the fourth postulate, that all right angles are equal. For an example of homogeneity, note that Euclid's proposition I.1 implies that the same equilateral triangle can be constructed at any location, not just in locations that are special in some way. The lack of boundaries follows from the second postulate, extensibility of a line segment.
Since propositions I.1-28 of the Elements represent every useful result that Euclid was able to prove without the use of the parallel postulate, all later geometrical propositions in the Elements should typically be expected to fail in elliptic geometry, becoming good approximations only on small distance scales.
For example, one way in which elliptic geometry differs from euclidean geometry is that the sum of the interior angles of a triangle is greater than 180 degrees. In the spherical model, for example, a triangle can be constructed with vertices at the locations where the three positive Cartesian coordinate axes intersect the sphere, and all three of its internal angles are 90 degrees, summing to 270 degrees. For sufficiently small triangles, the excess over 180 degrees can be made as small as desired.
The Pythagorean theorem fails in elliptic geometry. In the 90-90-90 triangle described above, all three sides have the same length, and they therefore do not satisfy . The Pythagorean result is recovered in the limit of small triangles.
The ratio of a circle's circumference to its area is smaller than in Euclidean geometry. In general, area and volume do not scale as the second and third powers of linear dimensions.
Higher-dimensional spaces
Hyperspherical model
The hyperspherical model is the generalization of the spherical model to higher dimensions. The points of n-dimensional elliptic space are the unit vectors in Rn+1, that is, the points on the surface of the unit ball in n+1 dimensional space. Lines in this model are great circles, intersections of the ball with flat hypersurfaces of dimension n passing through the origin.
Projective model
In the projective model, the points of n-dimensional real projective space are used as points of the model. The points of n-dimensional projective space can be identified with lines through the origin in (n+1)-dimensional space, and can be represented non-uniquely by nonzero vectors in Rn+1, with the understanding that u and ?u, for any non-zero scalar ?, represent the same point. Distance can be defined using the metric
This is homogeneous in each variable, with d(?u, µv) = d(u, v) if ? and µ are non-zero scalars, and so it defines a distance on the points of projective space.
A notable property of the projective model is that for even dimensions, such as the plane, the geometry is nonorientable, erasing the distinction between clockwise and counterclockwise rotation by identifying them.
Stereographic model
A model representing the same space as the hyperspherical model can be obtained by means of stereographic projection. Let En represent Rn ? , that is, n-dimensional real space extended by a single point at infinity. We may define a metric, the chordal metric, on
En by
where u and v are any two vectors in Rn and ||*|| is the usual Euclidean norm. We also define
The result is a metric space on En, which represents the distance along a chord of the corresponding points on the hyperspherical model,
which it maps bijectively to by stereographic projection. To obtain a model of elliptic geometry, we define another metric
The result is a model of elliptic geometry.
Self-consistency
Because elliptic geometry can be modeled, for example, as a spherical subspace of a Euclidean space, it follows that if Euclidean geometry is self-consistent, so is elliptic geometry. Therefore it is not possible to prove the parallel postulate based on the other four postulates of Euclidean geometry. Tarski proved that elementary Euclidean geometry is complete in a certain sense: there is an algorithm which, for every proposition, can show it to be either true or false. (This does not not violate Gödel's theorem, because Euclidean geometry cannot describe a sufficient amount of arithmetic for the theorem to apply.) It therefore follows that elementary elliptic geometry is also self-consistent and complete.
|
| |
|
|
